

<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
  <meta charset="utf-8">
  
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="Description" content="scikit-learn: machine learning in Python">

  
  <title>Lasso model selection: Cross-Validation / AIC / BIC &mdash; scikit-learn 0.22 documentation</title>
  
  <link rel="canonical" href="http://scikit-learn.org/stable/auto_examples/linear_model/plot_lasso_model_selection.html" />

  
  <link rel="shortcut icon" href="../../_static/favicon.ico"/>
  

  <link rel="stylesheet" href="../../_static/css/vendor/bootstrap.min.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/gallery.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
<script src="../../_static/jquery.js"></script> 
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
  <div class="container-fluid sk-docs-container px-0">
      <a class="navbar-brand py-0" href="../../index.html">
        <img
          class="sk-brand-img"
          src="../../_static/scikit-learn-logo-small.png"
          alt="logo"/>
      </a>
    <button
      id="sk-navbar-toggler"
      class="navbar-toggler"
      type="button"
      data-toggle="collapse"
      data-target="#navbarSupportedContent"
      aria-controls="navbarSupportedContent"
      aria-expanded="false"
      aria-label="Toggle navigation"
    >
      <span class="navbar-toggler-icon"></span>
    </button>

    <div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
      <ul class="navbar-nav mr-auto">
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../install.html">Install</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../user_guide.html">User Guide</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../modules/classes.html">API</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../index.html">Examples</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../getting_started.html">Getting Started</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../tutorial/index.html">Tutorial</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../glossary.html">Glossary</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../developers/index.html">Development</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../faq.html">FAQ</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../related_projects.html">Related packages</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../roadmap.html">Roadmap</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../about.html">About us</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
        </li>
        <li class="nav-item dropdown nav-more-item-dropdown">
          <a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
          <div class="dropdown-menu" aria-labelledby="navbarDropdown">
              <a class="sk-nav-dropdown-item dropdown-item" href="../../getting_started.html">Getting Started</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../tutorial/index.html">Tutorial</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../glossary.html">Glossary</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../developers/index.html">Development</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../faq.html">FAQ</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../related_projects.html">Related packages</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../roadmap.html">Roadmap</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../about.html">About us</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
          </div>
        </li>
      </ul>
      <div id="searchbox" role="search">
          <div class="searchformwrapper">
          <form class="search" action="../../search.html" method="get">
            <input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
            <input class="sk-search-text-btn" type="submit" value="Go" />
          </form>
          </div>
      </div>
    </div>
  </div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
    <input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
    <label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
    <div id="sk-sidebar-wrapper" class="border-right">
      <div class="sk-sidebar-toc-wrapper">
        <div class="sk-sidebar-toc-logo">
          <a href="../../index.html">
            <img
              class="sk-brand-img"
              src="../../_static/scikit-learn-logo-small.png"
              alt="logo"/>
          </a>
        </div>
        <div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
            <a href="plot_bayesian_ridge.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Bayesian Ridge Regression">Prev</a><a href="../index.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Examples">Up</a>
            <a href="plot_sparse_logistic_regression_20newsgroups.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Multiclass sparse logisitic regression on newgroups20">Next</a>
        </div>
        <div class="alert alert-danger p-1 mb-2" role="alert">
          <p class="text-center mb-0">
          <strong>scikit-learn 0.22</strong><br/>
          <a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
          </p>
        </div>
        <div class="alert alert-warning p-1 mb-2" role="alert">
          <p class="text-center mb-0">
            Please <a class="font-weight-bold" href="../../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
          </p>
        </div>
          <div class="sk-sidebar-toc">
            <ul>
<li><a class="reference internal" href="#">Lasso model selection: Cross-Validation / AIC / BIC</a></li>
</ul>

          </div>
      </div>
    </div>
    <div id="sk-page-content-wrapper">
      <div class="sk-page-content container-fluid body px-md-3" role="main">
        
  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-linear-model-plot-lasso-model-selection-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
</div>
<div class="sphx-glr-example-title section" id="lasso-model-selection-cross-validation-aic-bic">
<span id="sphx-glr-auto-examples-linear-model-plot-lasso-model-selection-py"></span><h1>Lasso model selection: Cross-Validation / AIC / BIC<a class="headerlink" href="#lasso-model-selection-cross-validation-aic-bic" title="Permalink to this headline">¶</a></h1>
<p>Use the Akaike information criterion (AIC), the Bayes Information
criterion (BIC) and cross-validation to select an optimal value
of the regularization parameter alpha of the <a class="reference internal" href="../../modules/linear_model.html#lasso"><span class="std std-ref">Lasso</span></a> estimator.</p>
<p>Results obtained with LassoLarsIC are based on AIC/BIC criteria.</p>
<p>Information-criterion based model selection is very fast, but it
relies on a proper estimation of degrees of freedom, are
derived for large samples (asymptotic results) and assume the model
is correct, i.e. that the data are actually generated by this model.
They also tend to break when the problem is badly conditioned
(more features than samples).</p>
<p>For cross-validation, we use 20-fold with 2 algorithms to compute the
Lasso path: coordinate descent, as implemented by the LassoCV class, and
Lars (least angle regression) as implemented by the LassoLarsCV class.
Both algorithms give roughly the same results. They differ with regards
to their execution speed and sources of numerical errors.</p>
<p>Lars computes a path solution only for each kink in the path. As a
result, it is very efficient when there are only of few kinks, which is
the case if there are few features or samples. Also, it is able to
compute the full path without setting any meta parameter. On the
opposite, coordinate descent compute the path points on a pre-specified
grid (here we use the default). Thus it is more efficient if the number
of grid points is smaller than the number of kinks in the path. Such a
strategy can be interesting if the number of features is really large
and there are enough samples to select a large amount. In terms of
numerical errors, for heavily correlated variables, Lars will accumulate
more errors, while the coordinate descent algorithm will only sample the
path on a grid.</p>
<p>Note how the optimal value of alpha varies for each fold. This
illustrates why nested-cross validation is necessary when trying to
evaluate the performance of a method for which a parameter is chosen by
cross-validation: this choice of parameter may not be optimal for unseen
data.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Olivier Grisel, Gael Varoquaux, Alexandre Gramfort</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">time</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>

<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LassoCV</span><span class="p">,</span> <span class="n">LassoLarsCV</span><span class="p">,</span> <span class="n">LassoLarsIC</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>

<span class="c1"># This is to avoid division by zero while doing np.log10</span>
<span class="n">EPSILON</span> <span class="o">=</span> <span class="mf">1e-4</span>

<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_diabetes</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">X</span><span class="p">,</span> <span class="n">rng</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">14</span><span class="p">)]</span>  <span class="c1"># add some bad features</span>

<span class="c1"># normalize data as done by Lars to allow for comparison</span>
<span class="n">X</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># LassoLarsIC: least angle regression with BIC/AIC criterion</span>

<span class="n">model_bic</span> <span class="o">=</span> <span class="n">LassoLarsIC</span><span class="p">(</span><span class="n">criterion</span><span class="o">=</span><span class="s1">&#39;bic&#39;</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">model_bic</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">t_bic</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t1</span>
<span class="n">alpha_bic_</span> <span class="o">=</span> <span class="n">model_bic</span><span class="o">.</span><span class="n">alpha_</span>

<span class="n">model_aic</span> <span class="o">=</span> <span class="n">LassoLarsIC</span><span class="p">(</span><span class="n">criterion</span><span class="o">=</span><span class="s1">&#39;aic&#39;</span><span class="p">)</span>
<span class="n">model_aic</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">alpha_aic_</span> <span class="o">=</span> <span class="n">model_aic</span><span class="o">.</span><span class="n">alpha_</span>


<span class="k">def</span> <span class="nf">plot_ic_criterion</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">color</span><span class="p">):</span>
    <span class="n">alpha_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">alpha_</span> <span class="o">+</span> <span class="n">EPSILON</span>
    <span class="n">alphas_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">alphas_</span> <span class="o">+</span> <span class="n">EPSILON</span>
    <span class="n">criterion_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">criterion_</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">alphas_</span><span class="p">),</span> <span class="n">criterion_</span><span class="p">,</span> <span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span>
             <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;</span><span class="si">%s</span><span class="s1"> criterion&#39;</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">alpha_</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
                <span class="n">label</span><span class="o">=</span><span class="s1">&#39;alpha: </span><span class="si">%s</span><span class="s1"> estimate&#39;</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;-log(alpha)&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;criterion&#39;</span><span class="p">)</span>


<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plot_ic_criterion</span><span class="p">(</span><span class="n">model_aic</span><span class="p">,</span> <span class="s1">&#39;AIC&#39;</span><span class="p">,</span> <span class="s1">&#39;b&#39;</span><span class="p">)</span>
<span class="n">plot_ic_criterion</span><span class="p">(</span><span class="n">model_bic</span><span class="p">,</span> <span class="s1">&#39;BIC&#39;</span><span class="p">,</span> <span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Information-criterion for model selection (training time </span><span class="si">%.3f</span><span class="s1">s)&#39;</span>
          <span class="o">%</span> <span class="n">t_bic</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># LassoCV: coordinate descent</span>

<span class="c1"># Compute paths</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing regularization path using the coordinate descent lasso...&quot;</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">LassoCV</span><span class="p">(</span><span class="n">cv</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">t_lasso_cv</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t1</span>

<span class="c1"># Display results</span>
<span class="n">m_log_alphas</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">alphas_</span> <span class="o">+</span> <span class="n">EPSILON</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span> <span class="o">=</span> <span class="mi">2300</span><span class="p">,</span> <span class="mi">3800</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">m_log_alphas</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">mse_path_</span><span class="p">,</span> <span class="s1">&#39;:&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">m_log_alphas</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">mse_path_</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="s1">&#39;k&#39;</span><span class="p">,</span>
         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Average across the folds&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">alpha_</span> <span class="o">+</span> <span class="n">EPSILON</span><span class="p">),</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span>
            <span class="n">label</span><span class="o">=</span><span class="s1">&#39;alpha: CV estimate&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>

<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;-log(alpha)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Mean square error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Mean square error on each fold: coordinate descent &#39;</span>
          <span class="s1">&#39;(train time: </span><span class="si">%.2f</span><span class="s1">s)&#39;</span> <span class="o">%</span> <span class="n">t_lasso_cv</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># LassoLarsCV: least angle regression</span>

<span class="c1"># Compute paths</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Computing regularization path using the Lars lasso...&quot;</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">LassoLarsCV</span><span class="p">(</span><span class="n">cv</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">t_lasso_lars_cv</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">t1</span>

<span class="c1"># Display results</span>
<span class="n">m_log_alphas</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">cv_alphas_</span> <span class="o">+</span> <span class="n">EPSILON</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">m_log_alphas</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">mse_path_</span><span class="p">,</span> <span class="s1">&#39;:&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">m_log_alphas</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">mse_path_</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="s1">&#39;k&#39;</span><span class="p">,</span>
         <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Average across the folds&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">alpha_</span><span class="p">),</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span>
            <span class="n">label</span><span class="o">=</span><span class="s1">&#39;alpha CV&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>

<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;-log(alpha)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Mean square error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Mean square error on each fold: Lars (train time: </span><span class="si">%.2f</span><span class="s1">s)&#39;</span>
          <span class="o">%</span> <span class="n">t_lasso_lars_cv</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes  0.000 seconds)</p>
<div class="sphx-glr-footer class sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-linear-model-plot-lasso-model-selection-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.22.X?urlpath=lab/tree/notebooks/auto_examples/linear_model/plot_lasso_model_selection.ipynb"><img alt="https://mybinder.org/badge_logo.svg" src="https://mybinder.org/badge_logo.svg" width="150px" /></a>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/97de47fe6488040c86acdcb8c97252e3/plot_lasso_model_selection.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_lasso_model_selection.py</span></code></a></p>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/a42feb850617e30a27dc22d020425c75/plot_lasso_model_selection.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_lasso_model_selection.ipynb</span></code></a></p>
</div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</div>


      </div>
    <div class="container">
      <footer class="sk-content-footer">
            &copy; 2007 - 2019, scikit-learn developers (BSD License).
          <a href="../../_sources/auto_examples/linear_model/plot_lasso_model_selection.rst.txt" rel="nofollow">Show this page source</a>
      </footer>
    </div>
  </div>
</div>
<script src="../../_static/js/vendor/bootstrap.min.js"></script>

<script>
    window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
    ga('create', 'UA-22606712-2', 'auto');
    ga('set', 'anonymizeIp', true);
    ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>


<script>
$(document).ready(function() {
    /* Add a [>>>] button on the top-right corner of code samples to hide
     * the >>> and ... prompts and the output and thus make the code
     * copyable. */
    var div = $('.highlight-python .highlight,' +
                '.highlight-python3 .highlight,' +
                '.highlight-pycon .highlight,' +
		'.highlight-default .highlight')
    var pre = div.find('pre');

    // get the styles from the current theme
    pre.parent().parent().css('position', 'relative');
    var hide_text = 'Hide prompts and outputs';
    var show_text = 'Show prompts and outputs';

    // create and add the button to all the code blocks that contain >>>
    div.each(function(index) {
        var jthis = $(this);
        if (jthis.find('.gp').length > 0) {
            var button = $('<span class="copybutton">&gt;&gt;&gt;</span>');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
            jthis.prepend(button);
        }
        // tracebacks (.gt) contain bare text elements that need to be
        // wrapped in a span to work with .nextUntil() (see later)
        jthis.find('pre:has(.gt)').contents().filter(function() {
            return ((this.nodeType == 3) && (this.data.trim().length > 0));
        }).wrap('<span>');
    });

    // define the behavior of the button when it's clicked
    $('.copybutton').click(function(e){
        e.preventDefault();
        var button = $(this);
        if (button.data('hidden') === 'false') {
            // hide the code output
            button.parent().find('.go, .gp, .gt').hide();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'hidden');
            button.css('text-decoration', 'line-through');
            button.attr('title', show_text);
            button.data('hidden', 'true');
        } else {
            // show the code output
            button.parent().find('.go, .gp, .gt').show();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'visible');
            button.css('text-decoration', 'none');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
        }
    });

	/*** Add permalink buttons next to glossary terms ***/
	$('dl.glossary > dt[id]').append(function() {
		return ('<a class="headerlink" href="#' +
			    this.getAttribute('id') +
			    '" title="Permalink to this term">¶</a>');
	});
  /*** Hide navbar when scrolling down ***/
  // Returns true when headerlink target matches hash in url
  (function() {
    hashTargetOnTop = function() {
        var hash = window.location.hash;
        if ( hash.length < 2 ) { return false; }

        var target = document.getElementById( hash.slice(1) );
        if ( target === null ) { return false; }

        var top = target.getBoundingClientRect().top;
        return (top < 2) && (top > -2);
    };

    // Hide navbar on load if hash target is on top
    var navBar = document.getElementById("navbar");
    var navBarToggler = document.getElementById("sk-navbar-toggler");
    var navBarHeightHidden = "-" + navBar.getBoundingClientRect().height + "px";
    var $window = $(window);

    hideNavBar = function() {
        navBar.style.top = navBarHeightHidden;
    };

    showNavBar = function() {
        navBar.style.top = "0";
    }

    if (hashTargetOnTop()) {
        hideNavBar()
    }

    var prevScrollpos = window.pageYOffset;
    hideOnScroll = function(lastScrollTop) {
        if (($window.width() < 768) && (navBarToggler.getAttribute("aria-expanded") === 'true')) {
            return;
        }
        if (lastScrollTop > 2 && (prevScrollpos <= lastScrollTop) || hashTargetOnTop()){
            hideNavBar()
        } else {
            showNavBar()
        }
        prevScrollpos = lastScrollTop;
    };

    /*** high preformance scroll event listener***/
    var raf = window.requestAnimationFrame ||
        window.webkitRequestAnimationFrame ||
        window.mozRequestAnimationFrame ||
        window.msRequestAnimationFrame ||
        window.oRequestAnimationFrame;
    var lastScrollTop = $window.scrollTop();

    if (raf) {
        loop();
    }

    function loop() {
        var scrollTop = $window.scrollTop();
        if (lastScrollTop === scrollTop) {
            raf(loop);
            return;
        } else {
            lastScrollTop = scrollTop;
            hideOnScroll(lastScrollTop);
            raf(loop);
        }
    }
  })();
});

</script>
    
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
    
</body>
</html>